Question

Let a and 8 be the zeros of g(2) = px²+ 4x+ 4. If a²+ b² = 24, what are the two possible values of p?​

Answer 1

Answer:

p = -1 , 2/3

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros .

★ A quadratic polynomial can have atmost two zeros .

★ If a and b are the zeros of the quadratic polynomial Ax² + Bx + C , then ;

Sum of zeros , (a + b) = -B/A

Product of zeros , (ab) = C/A

Solution:

Here,

The given quadratic polynomial is ;

g(x) = px² + 4x + 4

On comparing with the general form of quadratic polynomial Ax² + Bx + C , we have ;

A = p

B = 4

C = 4

Also,

a and b are the zeros of the given quadratic polynomial .

Thus,

=> Sum of zeros = -B/A

=> a + b = -4/p

Also,

=> Product of zeros = C/A

=> ab = 4/p

Now,

=> a + b = -4/p

=> (a + b)² = (-4/p)²

=> a² + b² + 2ab = 16/p²

=> 24 + 2(4/p) = 16/p²

=> 24 + 8/p = 16/p²

=> 24 + 8/p - 16/p² = 0

=> (24p² + 8p - 16) = 0

=> 24p² + 8p - 16 = 0

=> 24p² + 24p - 16p - 16 = 0

=> 24p(p + 1) - 16(p + 1) = 0

=> (p + 1)(24p - 16) = 0

=> p = -1 , p = 16/24

=> p = -1 , p = 2/3

Hence,

p = -1 , 2/3